Section 2.1.3.5.1. Accidental degeneracy

The symmetry analysis of lattice vibrations provides a powerful tool not only for the characterization of eigenvectors but also for the presentation of experimental results. In neutron scattering experiments, for example, a series of single phonons may be detected but symmetry determines which of these phonons belong to the same branch, i.e. how the single phonons have to be connected by a dispersion curve. The decision as to which of Figs. 2.1.3.8(a) or (b), which represent the same experimental results as full circles, is the correct one can be made by symmetry arguments only. In Fig. 2.1.3.8(a) the two phonon branches intersect. Thus, there are two degenerate phonons at the single wavevector . From the symmetry point of view, this particular wavevector has no special properties, i.e. the point group is just the same as for neighbouring wavevectors. Hence, the degeneracy cannot be due to symmetry and the respective eigenvectors and are not related by any transformation matrix. As a consequence, the two phonons cannot belong to the same irreducible representation, because otherwise every linear combination would equally well represent a valid eigenvector to the same eigenvalue with the same symmetry. It is, however, highly improbable that the special nature of the interatomic interactions gives rise to this uncertainty of eigenvectors at some wavevector within the Brillouin zone. Rather, it is expected that any infinitesimal change of force constants will favour one particular linear combination which, consequently, must correspond to a phonon of lower frequency. At the same time, there will be another well defined orthogonal eigenvector with slightly higher frequency – just as is represented in Fig. 2.1.3.8(b). Hence, two phonon branches of the same symmetry do not intersect and yield a frequency gap. This phenomenon is sometimes called anticrossing behaviour. It is associated with an eigenvector exchange between the two branches.

Figure 2.1.3.8 | top | pdf | (a) Accidental degeneracy of phonons with different symmetry. (b) Anticrossing of phonons with the same symmetry (no degeneracy).

Accidental degeneracy according to Fig. 2.1.3.8(a), on the other hand, can only be observed if the two phonon branches belong to different irreducible representations. In this case, the eigenvectors are uniquely determined even at since a mixing is forbidden by symmetry.